# What is the derivative of this function sin (x/2)?

$\frac{1}{2} \cos \left(\frac{x}{2}\right)$ apply the chain rule and the derivatives of trig functions.
$F \left(x\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$ is the chain rule and the derivative of $\sin x$ is $\cos x$ but we have a more complicated function inside the function, hence the chain rule.
$F \left(x\right) = \cos \left(\frac{x}{2}\right) \cdot \frac{d}{\mathrm{dx}} \left(\frac{1}{2} \cdot x\right)$